Abstract
It is shown, in particular, that if λn ≠ λk when n ≠ k, Re λn > 0, and\(\sum\limits_{n = 1}^\infty { (1 + \operatorname{Re} \lambda _n )} /|\lambda {}_n|^2< \infty\), then an entire function F that is bounded on the real line and represented by a Dirichlet series\(F(z) = \sum\limits_{n = 1}^\infty {}\) dn exp (λnz) that is uniformly and absolutely convergent on each compactum in ℂ is identically zero.
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Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 7, pp. 882–888, July, 1990.
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Vinnitskii, B.V., Shapovalovskii, A.V. Behavior on the real line of entire functions represented by Dirichlet series with complex exponents. Ukr Math J 42, 779–784 (1990). https://doi.org/10.1007/BF01062079
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DOI: https://doi.org/10.1007/BF01062079